Complexity Theory
Aus International Center for Computational Logic
Complexity Theory
Lehrveranstaltung mit SWS 4/2/0 (Vorlesung/Übung/Praktikum) im WS 2025
Dozent
Umfang (SWS)
- 4/2/0
Module
Leistungskontrolle
- Mündliche Prüfung
Vorlesungsreihe
Contents
This course covers both fundamental concepts and advanced topics in complexity theory.
Key topics are:
- Turing Machines (revision): Definition of Turing Machines; Variants; Computational Equivalence; Decidability and Recognizability; Enumeration
- Undecidability: Examples of Undecidable Problems; Mapping Reductions; Rice’s Theorem (both for characterizing Decidability and Recognizability); Recursion Theorem; Outlook into Decidability in Logic
- Time Complexity: Measuring Time Complexity; Many-One Reductions; Cook-Levin Theorem; Time Complexity Classes (P, NP, ExpTime); NP-completeness; pseudo-NP-complete problems
- Space Complexity: Space Complexity Classes (PSpace, L, NL); Savitch’s Theorem; PSpace-completeness; NL-completeness; NL = coNL
- Diagonalization: Hierarchy Theorems (det. Time, non-det. Time, Space); Gap Theorem; Ladner’s Theorem; Relativization; Baker-Gill-Solovay Theorem
- Alternation: Alternating Turing Machines; APTime = PSpace; APSpace = ExpTime; Polynomial Hierarchy
- Circuit Complexity: Boolean Circuits; Alternative Proof of Cook-Levin Theorem; Parallel Computation (NC); P-completeness; P/poly; (Karp-Lipton Theorem, Meyer’s Theorem)
- Probabilistic Computation: Randomized Complexity Classes (RP, PP, BPP, ZPP); Sipser-Gács-Lautemann Theorem
- Quantum Computing: Quantum circuits, BQP, some basic results
Mode of Teaching and Registration
The course generally does not require a special registration, and there is no participant limit. However, students in programmes that use the Selma system (esp., students in CMS Master) will need to register there to obtain credits. Most of the materials will be freely available worldwide.
Schedule and Location
This page will publish all dates (see Dates & Materials above).
- The weekly lecture sessions will take place on Mondays DS4 (13.00 - 14.30) in APB E005 and Tuesdays DS3 (11.10 - 12.40) in MER 0E23.
- The weekly exercise session will take place on Tuesday DS5 (14.50 - 16.20), also in APB E005.
Contact
Besides the regular meetings in lectures and exercise classes, you can also contact the teachers and other students in the public discussion channel on Matrix shown on the side.
Acknowledgements
The slides for some of the foundational lectures of this course are based on slides used by Markus Krötzsch for the course Complexity Theory at the University of Oxford, which were adopted from slides created by Stefan Kreutzer and Ian Horrocks for that course.
Further material has been prepared first by Daniel Borchmann during his time at TU Dresden.- Michael Sipser: Introduction to the Theory of Computation, International Edition; 3rd Edition; Cengage Learning 2013
- Introductory text that covers all basic topics in this lecture.
- Erich Grädel: Complexity Theory; Lecture Notes, Winter Term 2009/10. Available online at https://logic.rwth-aachen.de/Teaching/KTQC-WS09/index.html.en
- Free lecture notes with a general overview of main results; more detailed than Sipser on oracles and alternation; main reference for randomized computation
- John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation; Addison Wesley Publishing Company 1979
- The Cinderella Book; contains a lot of information not contained in most other books; the hierarchy of undecidable problems as well as Rice' characterization of recognizable properties of recognizable languages are from here.
- Christos H. Papadimitriou: Computational Complexity; 1995 Addison-Wesley Publishing Company, Inc
- Standard reference text for many advanced aspects on complexity theory; the proofs of the Linear Speedup Theorem, the Gap Theorem, and Ladner's Theorem as given in the lecture are from here
- Sanjeev Arora and Boaz Barak: Computational Complexity: A Modern Approach; Cambridge University Press 2009
- Extensive book covering the state of the art of Complexity Theory
- Michael R. Garey and David S. Johnson: Computers and Intractability; Bell Telephone Laboratories, Inc. 1979
- The classical book on Complexity Theory; contains a long list of problems with their complexities
Veranstaltungskalender abonnieren (icalendar)
| Vorlesung | Introduction and Motivation | DS4, 13. Oktober 2025 in APB E005 | Datei 1, Datei 2 |
| Vorlesung | Turing Machines and Languages | DS3, 14. Oktober 2025 in MER 0E23 | Datei 1, Datei 2 |
| Vorlesung | Undecidability | DS4, 20. Oktober 2025 in APB E005 | Datei 1, Datei 2 |
| Vorlesung | Undecidability and Recursion | DS3, 21. Oktober 2025 in MER 0E23 | Datei 1, Datei 2 |
| Übung | Mathematical Foundations, Decidability, and Recognisability | DS5, 21. Oktober 2025 in APB E005 | Datei |
| Vorlesung | Time Complexity and Polynomial Time | DS4, 27. Oktober 2025 in APB E005 | Datei 1, Datei 2 |
| Vorlesung | Polynomial-Time Reductions and Nondeterministic Polynomial Time | DS3, 28. Oktober 2025 in MER 0E23 | Datei 1, Datei 2 |
| Übung | Undecidability | DS5, 28. Oktober 2025 in APB E005 | Datei |
| Vorlesung | NP-Completeness | DS4, 3. November 2025 in APB E005 | Datei 1, Datei 2 |
| Vorlesung | NP-Complete Problems | DS3, 4. November 2025 in MER 0E23 | Datei 1, Datei 2 |
| Übung | Time Complexity | DS5, 4. November 2025 in APB E005 | Datei |
| Vorlesung | Space Complexity and PSPACE | DS4, 10. November 2025 in APB E005 | Datei 1, Datei 2 |
| Vorlesung | Polynomial Space and Games | DS3, 11. November 2025 in MER 0E23 | Datei 1, Datei 2 |
| Übung | NP-Completeness | DS5, 11. November 2025 in APB E005 | Datei |
| Vorlesung | Logarithmic Space | DS4, 17. November 2025 in APB E005 | Datei 1, Datei 2 |
| Vorlesung | Hierarchy Theorems | DS3, 18. November 2025 in MER 0E23 | Datei 1, Datei 2 |
| Übung | More NP-Completeness and P-Completeness | DS5, 18. November 2025 in APB E005 | |
| Vorlesung | Space Hierarchy and Gaps | DS4, 24. November 2025 in APB E005 | Datei 1, Datei 2 |
| Vorlesung | P vs. NP: Ladner's Theorem | DS3, 25. November 2025 in MER 0E23 | Datei 1, Datei 2 |
| Übung | Space Complexity | DS5, 25. November 2025 in APB E005 | Datei |
| Vorlesung | P vs. NP and Diagonalisation | DS4, 1. Dezember 2025 in APB E005 | Datei 1, Datei 2 |
| Vorlesung | Alternation | DS3, 2. Dezember 2025 in MER 0E23 | Datei 1, Datei 2 |
| Übung | Time and Space Complexity | DS5, 2. Dezember 2025 in APB E005 | Datei |
| Vorlesung | The Polynomial Hierarchy | DS4, 8. Dezember 2025 in APB E005 | |
| Vorlesung | Polynomial Hierarchy / Circuit Complexity | DS3, 9. Dezember 2025 in MER 0E23 | |
| Übung | Alternation | DS5, 9. Dezember 2025 in APB E005 | |
| Vorlesung | Circuits and Parallel Computation | DS4, 15. Dezember 2025 in APB E005 | |
| Vorlesung | Questions and Answers (before Christmas) | DS3, 16. Dezember 2025 in MER 0E23 | |
| Entfällt | Christmas Break | DS5, 16. Dezember 2025 in APB E005 | |
| Vorlesung | Probabilistic Turing Machines | DS4, 5. Januar 2026 in APB E005 | |
| Vorlesung | Probabilistic Complexity Classes (1) | DS3, 6. Januar 2026 in MER 0E23 | |
| Übung | Polynomial Hierarchy | DS5, 6. Januar 2026 in APB E005 | |
| Vorlesung | Probabilistic Complexity Classes (2) | DS4, 12. Januar 2026 in APB E005 | |
| Vorlesung | Quantum Computing (1) | DS3, 13. Januar 2026 in MER 0E23 | |
| Übung | Circuit Complexity | DS5, 13. Januar 2026 in APB E005 | |
| Vorlesung | Quantum Computing (2) | DS4, 19. Januar 2026 in APB E005 | |
| Vorlesung | Interactive Proof Systems (1) | DS3, 20. Januar 2026 in MER 0E23 | |
| Übung | Randomised Computation | DS5, 20. Januar 2026 in APB E005 | |
| Vorlesung | TBA | DS4, 26. Januar 2026 in APB E005 | |
| Vorlesung | TBA | DS3, 27. Januar 2026 in MER 0E23 | |
| Übung | Quantum Computing | DS5, 27. Januar 2026 in APB E005 | |
| Vorlesung | TBA | DS4, 2. Februar 2026 in APB E005 | |
| Vorlesung | Summary and Consultation | DS3, 3. Februar 2026 in MER 0E23 | |
| Übung | TBA | DS5, 3. Februar 2026 in APB E005 |
Kalender
